Flow through a hole in a sphere

Question

I have a sphere of diameter $D$ (radius $R$) with a small hole of diameter $d$ (radius $a$) in it, with air flowing through the axis of the hole:

I am trying to estimate the drag force due to friction in the hole.

First let's consider simplifying assumptions:

1. $d << D$
2. Pressure distribution around the sphere is unaffected by the small hole.
3. $Re_D = \frac{\rho u_{\infty}D}{\mu} >> 1 \therefore$ inviscid flow around sphere.
4. $Re_d\frac{d}{D} << 1 \therefore$ inertia free flow in the hole via Navier-Stokes scaling.

Assumptions 2 and 3 suggest that pressures at points 1 and 2 (shown in the figure) can be estimated via simple stagnation pressures with Bernoulli's equation:

$$ P_1 = P_2 = P_{\infty} + \frac{1}{2} \rho u_{\infty}^2 $$

Now, assumptions 1 and 4 suggest that the flow in the hole is viscous dominated, thus yielding the Hagen-Poiseuille solution for flow through a pipe:

$$v_z(r) = \frac{1}{4\mu}\frac{\Delta P}{D}(a^2 - r^2) $$

From this, I could easily calculate the shear stress in the hole, and therefore the drag force in the hole.

The problem, however, is that my Bernoulli analysis yields a zero pressure drop:

$$\Delta P = P_1 - P_2 = 0$$

This seems to suggest that there is no flow through the small hole in an inviscid regime, where $Re_D >> 1$.

In that case, I have zero friction drag in the hole since there is approximately no flow.

Is this analysis correct? Would there be approximately no flow and therefore no friction drag in this small hole, if there is large Reynolds number flow around the sphere?

Answer 1

It looks like my issue of $\Delta P = 0$ is a case of D'Alembert's paradox, where purely inviscid flow wrongfully predicts zero pressure drop around an object. Furthermore, if this is high Reynolds number flow $(Re_D >> 1 )$, it is most likely in the turbulent regime, and experiments have shown that that $\Delta P \sim 0.5\rho u_{\infty}^2$, as shown below with the coefficient of pressure $C_P$.

This result was suggested by user Deep in a comment on the OP. Here I will attempt an argument to show this result analytically.

My analysis for the stagnation pressure at point 1 is fine:

$$ P_1 = P_{\infty} + 0.5\rho u_{\infty}^2$$

Now we can consider the drag force and some simple scaling to arrive at $P_2$.

For a sphere in the high $Re_D$ regime, we know that the drag coefficient $C_D = 0.4$, thus the drag force is:

$$ F_D = 0.4(0.5\rho u_{\infty}^2)\pi R^2 = 0.628\rho u_{\infty}^2R^2$$

And $F_D \sim R^2\Delta P$, therefore:

$$ \Delta P = P_1 - P_2 \sim 0.628\rho u_{\infty}^2$$

Solving for $P_2$,

$$ P_2 \sim P_1 - 0.628\rho u_{\infty}^2 = P_{\infty} - 0.128\rho u_{\infty}^2$$

where we can ignore the $0.128\rho u_{\infty}^2$ term? If we do ignore it, the result is:

$$ P_2 \sim P_{\infty}$$

This seems kind of hand-wavy, but it's the closest I've come to convincing myself with analytical arguments that $ P_2 \sim P_{\infty}$. Does anyone know of a better analysis?